Question 23.12: Show that the functions cos mωt and cos nωt with m, n positi......
Show that the functions cos mωt and cos nωt with m, n positive integers and m ≠ n are orthogonal on the interval -\frac{\pi}{\omega} \leqslant t \leqslant \frac{\pi}{\omega}.
Step-by-Step
We must evaluate
\int_{-\pi / \omega}^{\pi / \omega} \cos m \omega t \cos n \omega t \mathrm{~d} tUsing the trigonometric identity 2 cos A cos B = cos(A + B) +cos(/A – B), we find the integral becomes
\begin{aligned} & \frac{1}{2} \int_{-\pi / \omega}^{\pi / \omega} \cos (m+n) \omega t+\cos (m-n) \omega t \mathrm{~d} t \\ & =\frac{1}{2}\left[\frac{\sin (m+n) \omega t}{(m+n) \omega}+\frac{\sin (m-n) \omega t}{(m-n) \omega}\right]_{-\pi / \omega}^{\pi / \omega} \\ & =0 \end{aligned}since sin(m ± n)π = 0 for all integers m, n. It was necessary to require m ≠ n since otherwise the second quantity in brackets becomes undefined. Hence cos mωt and cos nωt are orthogonal on the given interval.
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